The relationship between the GDP gap and the recessionary gap is that the recessionary gap is part of the GDP gap.
A GDP gap can be defined as the gap between what the economy could be producing at full employment and what it actually is producing. In a recession, the economy does not produce as much as it could. The gap between what it could produce and what it is producing is called the GDP gap.
In Keynesian economics, the government is supposed to spend money to increase aggregate demand and close the GDP gap. The amount of money that it needs to spend to do this is known as the recessionary gap. The recessionary gap is not exactly equal to the GDP gap. This is because of the multiplier effect. When the government spends a given amount of money, its effect on GDP is greater than the amount of money that it spent. This is because the money is spent over and over again as people who receive the money use it to buy goods and services.
Therefore, the GDP gap is equal to the recessionary gap multiplied by the multiplier. This is the relationship between the two.